# A triangle has sides A,B, and C. If the angle between sides A and B is (3pi)/8, the angle between sides B and C is pi/4, and the length of B is 7, what is the area of the triangle?

Feb 22, 2016

≈ 17.32 square units

#### Explanation:

The area of a triangle can be calculated using $\frac{1}{2} a b \sin \theta$
where $\theta \text{ is the angle between a and b }$

In this triangle , only know the length of one side B. Require to find the length of A or C.

This can be done using the$\textcolor{b l u e}{\text{ sine rule }}$

$\frac{A}{\sin} A = \frac{B}{\sin} B = \frac{C}{\sin} C$
where the angles A , B and C on the denominator represent the angles opposite the corresponding sides A , B and C.

$\textcolor{red}{\text{ Calculating the length of side C}}$

using $\frac{B}{\sin} B = \frac{C}{\sin} C$

Before using this, require the size of angle B

The sum of the 3 angles in a triangle $= \pi$

angle B $= \left[\pi - \left(\frac{3 \pi}{8} + \frac{\pi}{4}\right)\right] = \pi - \frac{5 \pi}{8} = \frac{3 \pi}{8}$

Now angle B = angle C hence side B = sideC = 7

In this question B = C = 7 and $\theta = \frac{\pi}{4}$

"area" = 1/2xxBxxCsintheta = 1/2xx7xx7xxsin(pi/4) ≈ 17.32